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I heard a podcast about a study (I think it was Radiolab or similar) that found that people are viewed as more likable after they reveal a weakness or a shortcoming, but the effect only occurs if they have established that they are competent.

What is that study? The podcast said it is influential and well known.

In the podcast, a professor told a story about how when he was a graduate student made a fool of himself in front of his PhD advisor and he joked that he did it on purpose to improve his likebility and she retorted that he hadn't yet established himself as competent. I remember hearing it in late 2014. Any help enormously appreciated!

An experiment was performed which demonstrated that the attractiveness
of a superior person is enhanced if he commits a clumsy blunder; the
same blunder tends to decrease the the attractiveness of a mediocre
person. We predicted these results by conjecturing that a superior
person may be viewed as superhuman and, therefore, distant; a blunder
tends to humanize him and, consequently, increases his attractiveness.

$\begingroup$If GS is even somewhat correct, the 1966 paper only has 221 citations, which is pretty low for something this old and supposedly famous (I do see it mentioned in a few books though). Although GS tends to undeestimate for old papers, a 1959 paper of Aronson "The effect of severity of initiation on liking for a group." on effort justification is credited with 1300+ citations. Also, a pratfall is not the same thing as disclosing a weakness.$\endgroup$
– FizzJan 3 '18 at 1:35

I'm not convinced the effect is even true in general, and appears to be false in job/task-related contexts particularly when the boss does the disclosure; according to a recent paper:

It is generally believed that self-disclosure has positive effects, particularly for relationships; however, we predict
and find negative effects in the context of task-oriented relationships. Across three laboratory experiments, we
find that both task-relevant (Study 1) and task-irrelevant (Studies 2 and 3) weakness disclosures, made by a
higher (versus peer) status coworker during an interdependent task, negatively affected the receiver’s perception
of the discloser’s status and consequently undermined the discloser’s influence, encouraged task conflict, and led
to lower relationship quality with the discloser. Peer status disclosers did not trigger these negative responses.
We find support for perceived vulnerability as the proposed psychological process (Study 3). Specifically, higher
(but not peer) status disclosers experience a status penalty after weakness disclosures because these disclosures
signal vulnerability, which violates the expectations people have for higher (but not peer) status coworkers.
These findings provide insight into the effects of self-disclosing weakness at work and the ways in which high
status employees may inadvertently trigger their own status loss.

What this study (alas) didn't address is whether a low-status peer would be seen in a more positive light by a higher-status one after a disclosure, like in your last paragraph. So we don't seem to have experimental evidence on whether "sucking up" by (more or less machiavellian) disclosures or even faking less competence (as to diminish perceived threat) works as intended. I suppose the latter is quite contextual. In a hierarchy where the lower status individual could conceivably take the place of the higher status individual (say in a military junta or an apparatchik system) feigning a less threatening attitude (via some but not complete incompetence) perhaps does work. I'm having doubts that feigning partial incompetence does much good to a subordinate in a context where the high-status individual does not see any direct threat to his position (like when hiring a contractor for a non-core task). But this is just me theorizing. More empirical research is needed, as the saying goes. In general, other forms of "sucking up" do seem to work through, with the caveat that while "sucking up" may have a positive effect on the relationship with the boss, it's also likely to have a negative effect on the relationship(s) with same-status peers/colleagures.

$\begingroup$The exception to the rule (peer status disclosers) might be related to what the OP was hinting at? Unfortunately no time to read the full paper. :)$\endgroup$
– Steven Jeuris♦Dec 23 '17 at 18:40